openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationSpatial.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticScalarFourth | |
| namespace OAI | |
| /-! Three spatial derivatives of a bounded continuation through a single | |
| actual Gaussian transition. These are the derivatives needed to close the | |
| inverse-coordinate continuation at the two mean-spin endpoints. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| open scoped NNReal | |
| namespace InvariantIsing | |
| def magneticContinuationSecond (ζ : ℝ) (v : ℝ≥0) | |
| (F M Q A B C : ℝ → ℝ) (z : ℝ) : ℝ := | |
| let T := fieldSpinTransition ζ v F | |
| let D := fieldTiltSpatial ζ v F M | |
| D B C z + ζ * (D (fun y => A y * M y) (fun y => B y * M y + A y * Q y) z - | |
| (D A B z * T M z + T A z * D M Q z)) | |
| def magneticContinuationThird (ζ : ℝ) (v : ℝ≥0) | |
| (F M Q R A B C D : ℝ → ℝ) (z : ℝ) : ℝ := | |
| let T := fieldSpinTransition ζ v F | |
| let S := fieldTiltSpatial ζ v F M | |
| magneticContinuationSecond ζ v F M Q B C D z + ζ * | |
| (magneticContinuationSecond ζ v F M Q | |
| (fun y => A y * M y) (fun y => B y * M y + A y * Q y) | |
| (fun y => C y * M y + 2 * B y * Q y + A y * R y) z - | |
| (magneticContinuationSecond ζ v F M Q A B C z * T M z + | |
| 2 * S A B z * S M Q z + | |
| T A z * magneticContinuationSecond ζ v F M Q M Q R z)) | |
| lemma hasDerivAt_magneticContinuationSecond (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q A B C : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : Measurable M) (hQ : Measurable Q) | |
| (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) | |
| {K L a b c : ℝ} (hK : 0 ≤ K) (_hL : 0 ≤ L) (ha : 0 ≤ a) (hb : 0 ≤ b) | |
| (bM : ∀ y, |M y| ≤ K) (bQ : ∀ y, |Q y| ≤ L) | |
| (bA : ∀ y, |A y| ≤ a) (bB : ∀ y, |B y| ≤ b) (bC : ∀ y, |C y| ≤ c) | |
| (dF : ∀ y, HasDerivAt F (M y) y) (dM : ∀ y, HasDerivAt M (Q y) y) | |
| (dA : ∀ y, HasDerivAt A (B y) y) (dB : ∀ y, HasDerivAt B (C y) y) | |
| (z : ℝ) : | |
| HasDerivAt (fieldTiltSpatial ζ v F M A B) | |
| (magneticContinuationSecond ζ v F M Q A B C z) z := by | |
| have bAM : ∀ y, |A y * M y| ≤ a * K := by | |
| intro y | |
| rw [abs_mul] | |
| exact mul_le_mul (bA y) (bM y) (abs_nonneg _) ha | |
| have bAM' : ∀ y, |B y * M y + A y * Q y| ≤ b * K + a * L := by | |
| intro y | |
| refine (abs_add_le _ _).trans (add_le_add ?_ ?_) | |
| · rw [abs_mul] | |
| exact mul_le_mul (bB y) (bM y) (abs_nonneg _) hb | |
| · rw [abs_mul] | |
| exact mul_le_mul (bA y) (bQ y) (abs_nonneg _) ha | |
| have htA := hasDerivAt_fieldSpinTransition ζ v hF hG hM hA hB hK ha bM bA bB dF dA z | |
| have htB := hasDerivAt_fieldSpinTransition ζ v hF hG hM hB hC hK hb bM bB bC dF dB z | |
| have htM := hasDerivAt_fieldSpinTransition ζ v hF hG hM hM hQ hK hK bM bM bQ dF dM z | |
| have htAM := hasDerivAt_fieldSpinTransition ζ v hF hG hM (hA.mul hM) | |
| ((hB.mul hM).add (hA.mul hQ)) hK (mul_nonneg ha hK) bM bAM bAM' | |
| dF (fun y => (dA y).mul (dM y)) z | |
| exact htB.add ((htAM.sub (htA.mul htM)).const_mul ζ) | |
| lemma hasDerivAt_magneticContinuationThird (ζ : ℝ) (v : ℝ≥0) | |
| {F M Q R A B C D : ℝ → ℝ} (hF : Measurable F) (hG : HasLinearGrowth F) | |
| (hM : Measurable M) (hQ : Measurable Q) (hR : Measurable R) | |
| (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) (hD : Measurable D) | |
| {K L J a b c d : ℝ} (hK : 0 ≤ K) (hL : 0 ≤ L) | |
| (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) | |
| (bM : ∀ y, |M y| ≤ K) (bQ : ∀ y, |Q y| ≤ L) (bR : ∀ y, |R y| ≤ J) | |
| (bA : ∀ y, |A y| ≤ a) (bB : ∀ y, |B y| ≤ b) | |
| (bC : ∀ y, |C y| ≤ c) (bD : ∀ y, |D y| ≤ d) | |
| (dF : ∀ y, HasDerivAt F (M y) y) (dM : ∀ y, HasDerivAt M (Q y) y) | |
| (dQ : ∀ y, HasDerivAt Q (R y) y) | |
| (dA : ∀ y, HasDerivAt A (B y) y) (dB : ∀ y, HasDerivAt B (C y) y) | |
| (dC : ∀ y, HasDerivAt C (D y) y) (z : ℝ) : | |
| HasDerivAt (magneticContinuationSecond ζ v F M Q A B C) | |
| (magneticContinuationThird ζ v F M Q R A B C D z) z := by | |
| have bprod {U V : ℝ → ℝ} {u w : ℝ} (hu : 0 ≤ u) | |
| (bu : ∀ y, |U y| ≤ u) (bv : ∀ y, |V y| ≤ w) (y : ℝ) : | |
| |U y * V y| ≤ u * w := by | |
| rw [abs_mul] | |
| exact mul_le_mul (bu y) (bv y) (abs_nonneg _) hu | |
| have bAM := bprod ha bA bM | |
| have bAM' (y : ℝ) : |B y * M y + A y * Q y| ≤ b * K + a * L := | |
| (abs_add_le _ _).trans (add_le_add (bprod hb bB bM y) (bprod ha bA bQ y)) | |
| have bAM'' (y : ℝ) : |C y * M y + 2 * B y * Q y + A y * R y| ≤ | |
| c * K + 2 * b * L + a * J := by | |
| have h2 : |2 * B y * Q y| ≤ 2 * b * L := by | |
| rw [show 2 * B y * Q y = 2 * (B y * Q y) by ring, abs_mul] | |
| rw [abs_of_pos (by norm_num : (0 : ℝ) < 2)] | |
| simpa only [mul_assoc] using | |
| mul_le_mul_of_nonneg_left (bprod hb bB bQ y) (by norm_num : (0 : ℝ) ≤ 2) | |
| exact (abs_add_le _ _).trans (add_le_add | |
| ((abs_add_le _ _).trans (add_le_add (bprod hc bC bM y) h2)) (bprod ha bA bR y)) | |
| have dAM (y : ℝ) : HasDerivAt (fun x => A x * M x) | |
| (B y * M y + A y * Q y) y := (dA y).mul (dM y) | |
| have dAM' (y : ℝ) : HasDerivAt (fun x => B x * M x + A x * Q x) | |
| (C y * M y + 2 * B y * Q y + A y * R y) y := by | |
| convert ((dB y).mul (dM y)).add ((dA y).mul (dQ y)) using 1 | |
| ring | |
| have hSA := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hA hB hC | |
| hK hL ha hb bM bQ bA bB bC dF dM dA dB z | |
| have hSB := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hB hC hD | |
| hK hL hb hc bM bQ bB bC bD dF dM dB dC z | |
| have hSM := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ hM hQ hR | |
| hK hL hK hL bM bQ bM bQ bR dF dM dM dQ z | |
| have hSAM := hasDerivAt_magneticContinuationSecond ζ v hF hG hM hQ | |
| (hA.mul hM) ((hB.mul hM).add (hA.mul hQ)) | |
| (((hC.mul hM).add ((measurable_const.mul hB).mul hQ)).add (hA.mul hR)) | |
| hK hL (mul_nonneg ha hK) (by positivity) bM bQ bAM bAM' bAM'' | |
| dF dM dAM dAM' z | |
| change HasDerivAt (fieldTiltSpatial ζ v F M (fun y => A y * M y) | |
| (fun y => B y * M y + A y * Q y)) | |
| (magneticContinuationSecond ζ v F M Q (fun y => A y * M y) | |
| (fun y => B y * M y + A y * Q y) | |
| (fun y => C y * M y + 2 * B y * Q y + A y * R y) z) z at hSAM | |
| have hTA : HasDerivAt (fieldSpinTransition ζ v F A) | |
| (fieldTiltSpatial ζ v F M A B z) z := | |
| hasDerivAt_fieldSpinTransition ζ v hF hG hM hA hB hK ha bM bA bB dF dA z | |
| have hTM : HasDerivAt (fieldSpinTransition ζ v F M) | |
| (fieldTiltSpatial ζ v F M M Q z) z := | |
| hasDerivAt_fieldSpinTransition ζ v hF hG hM hM hQ hK hK bM bM bQ dF dM z | |
| have hh := hSB.add ((hSAM.sub ((hSA.mul hTM).add (hTA.mul hSM))).const_mul ζ) | |
| convert hh using 1 | |
| · rfl | |
| · dsimp only [magneticContinuationThird] | |
| ring | |
| end InvariantIsing | |
| end | |
| end OAI | |